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On the Realizability of the Critical Points of a Realizable List

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On the Realizability of the Critical Points of a Realizable List On the realizability of the critical points of a realizable list✩ a,1 b,1 c,1, Sarah L. Hoover , Daniel A. McCormick , Pietro Paparella ∗, Amber R. Thrallc,1 aHamilton College, 198 College Hill Rd., Clinton, NY, 13323, USA bFlorida Institute of Technology, 150 W. University Blvd., Melbourne, FL, 32901, USA cDivision of Engineering and Mathematics, University of Washington Bothell, 18115 Campus Way NE, Bothell, WA 98011, USA Abstract The nonnegative inverse eigenvalue problem (NIEP) is to characterize the spec- tra of entrywise nonnegative matrices. A finite multiset of complex numbers is called realizable if it is the spectrum of an entrywise nonnegative matrix. Monov conjectured that the kth-moments of the list of critical points of a real- izable list are nonnegative. Johnson further conjectured that the list of critical points must be realizable. In this work, Johnson’s conjecture, and consequently Monov’s conjecture, is established for a variety of important cases including Ciarlet spectra, Sule˘ımanova spectra, spectra realizable via companion matri- ces, and spectra realizable via similarity by a complex Hadamard matrix. Ad- ditionally we prove a result on differentiators and trace vectors, and use it to provide an alternate proof of a result due to Malamud and a generalization of a result due to Kushel and Tyaglov on circulant matrices. Implications for further research are discussed. Keywords: Nonnegative inverse eigenvalue problem, Sule˘ımanova spectrum, differentiator, trace vector, complex Hadamard matrix 2010 MSC: 15A29, 15A18, 15B48, 30C15 1. Introduction arXiv:1712.05454v1 [math.SP] 14 Dec 2017 The longstanding nonnegative inverse eigenvalue problem (NIEP) is to char- acterize the spectra of entrywise nonnegative matrices. More specifically, given a finite multi-set (herein list)Λ= λ ,...,λn of complex numbers, the NIEP { 1 } ✩This work was supported by NSF Award DMS-1460699. ∗Corresponding author. Email addresses: [email protected] (Sarah L. Hoover), [email protected] (Daniel A. McCormick), [email protected] (Pietro Paparella), [email protected] (Amber R. Thrall) URL: http://faculty.washington.edu/pietrop/ (Pietro Paparella), http://amber.thrall.me/ (Amber R. Thrall) Preprint submitted to Linear Algebra and its Applications September 18, 2018 asks for necessary and sufficient conditions such that Λ is the spectrum of an n-by-n entrywise-nonnegative matrix. If Λ is the spectrum of a nonnegative matrix A, then Λ is called realizable and A is a called realizing matrix (for Λ). It is well-known that if Λ = λ1,...,λn is realizable, then Λ must satisfy the following conditions: { } Λ := λ¯ ,..., λ¯n = Λ; (1) { 1 } ρ(Λ) := max λi Λ; (2) 1 i n ≤ ≤ {| |} ∈ n k sk(Λ) := λ 0, k N; and (3) i ≥ ∀ ∈ i=1 m Xm 1 s (Λ) n − skm(Λ), k,m N. (4) k ≤ ∀ ∈ It is worth noting that these conditions are not independent: Loewy and London [24] showed that the moment condition (3) implies the self-conjugacy condition (1); Friedland [12, Theorem 1] showed that the eventual nonnegativity of the moments implies the spectral radius condition (2); and if the trace is nonnega- tive, i.e., if s1(Λ) 0, then the J-LL condition (4) (established independently by Johnson [15] and≥ by Loewy and London [24]) implies the moment condition since 1 k sk(Λ) k 1 s1 (Λ) 0. ≥ n − ≥ Thus, the J-LL condition and the nonnegativity of the trace imply the self- conjugacy, spectral radius, and moment conditions. There is another necessary condition due to Holtz [13], which was shown to be independent of J-LL, but for brevity, it will not be considered here (for more information regarding the NIEP, there are several articles [2, 6, 11, 17, 26, 32] and monographs [26, 1] that discuss the problem). n Let Λ = λ ,...,λn C (n 2) be a list, let p(z) = (z λi), and { 1 } ⊂ ≥ i=1 − let Λ′ = µ1,...,µn 1 denote the zeros of p′. Monov [27] posed the following conjecture.{ − } Q Conjecture 1.1 (Monov). If Λ is realizable, then sk(Λ′) 0 for all k N. ≥ ∈ The following is due to Johnson [16]. Conjecture 1.2 (Johnson). If Λ is realizable, then Λ′ is realizable. It is clear that Johnson’s conjecture implies Monov’s conjecture. Cronin and Laffey [8] established Johnson’s conjecture in the cases when n 4, or when ≤ n 6 and s1(Λ) = 0, . ≤Note that the converse of Conjecture 1.2 is not true; consider the list Λ = 1, 1, 2/3, 2/3, 2/3 and the polynomial p(z) = (z 1)2(z +2/3)3. Then { − 4 − 2 − } − p′(z)=5z 5z 20/27z+20/27,Λ′ = 1, 1/3, 2/3, 2/3 ,andΛ′ is realizable [24]. However,− it− is well-known that Λ is{ not realizable− − (see,} e.g., Friedland [12] and Paparella and Taylor [28] for a more general result). Note that there are instances when the constant of integration can be selected so that the zeros of an antiderivative form a realizable set (see Section 6). 2 In this work, we investigate and constructively prove Conjecture 1.2 for a variety of cases. First, we examine the kth moments of the critical points of a realizable list and establish a sufficient condition for realizability via companion matrices. Next, we explore the d-companion matrix as a construction for a realizing matrix of the critical points and establish a sufficient condition that includes Ciarlet spectra [7]. We then study differentiators following the work of Pereira [29] and show that Johnson’s conjecture holds for spectra realizable via complex Hadamard similarities. Along the way, we provide an alternate proof to a result of Malamud [25] and extend the results on circulant matrices by Kushel and Tyaglov [20]. Additionally, we use these results to provide a new proof of a classical theorem on the interlacing roots and critical points for polynomials. Finally, implications for further research are discussed. 2. Notation n Unless otherwise stated, Λ := λ ,...,λn C (n 2), p(t) := (t { 1 } ⊂ ≥ i=1 − λi) C[t], and Λ′ := µ C p′(µ)=0 = µ1,...,µn 1 . Furthermore, we ∈ { ∈ | } { − } Q refer to the critical points of p, i.e., the zeros of p′, as the critical points of Λ. For n N, we let n denote the set 1,...,n . ∈ h i { } We let Mm n(F) denote the set of m-by-n matrices with entries from a field × F. In the case when m = n, Mm n(F) is abbreviated to Mn(F). × For A Mn(F), we let σ(A) denote the spectrum (i.e., list of eigenvalues) of A; we let ∈A denote the ith principal submatrix of A, i.e., A is the (n 1)- (i) (i) − by-(n 1) matrix obtained by deleting the ith-row and ith-column of A; and we let τ(A−) denote the normalized trace of A, i.e., τ(A) := (1/n) tr A. th We let diag (λ1,...,λn) denote the diagonal matrix whose i diagonal entry is λi. For A Mm(F) and B Mn(F), the direct sum of A and B, denoted A B, is the matrix∈ ∈ ⊕ A 0 Mm n(F). 0 B ∈ × The Kronecker product of A Mm n(F) and B Mp q(F), denoted A B, is the (mp)-by-(nq) block matrix∈ × ∈ × ⊗ a11B ... a1nB . . .. . a B ... a B m1 mn 3. Preliminary Results Proposition 3.1. If Λ is realizable, then Λ′ = Λ′. Proof. Follows from the fact that the derivative of a real polynomial is real. Proposition 3.2. If Λ is realizable, then s (Λ′) 0. 1 ≥ 3 Proof. Since 1 1 s (Λ) = s (Λ′), n 1 n 1 1 − it follows that n 1 s (Λ′)= − s (Λ) 0. 1 n 1 ≥ Monov [27, Proposition 2.3] provides a formula that relates the moments of a list with the moments of its critical points. Theorem 3.3 (Monov’s moment formula). If s1(Λ) n 0 ... 0 2s (Λ) s (Λ) n . 0 2 1 . dk(Λ) := . , (k 1)sk 1(Λ) sk 2(Λ) sk 3(Λ) ... n − − − − ksk(Λ) sk 1(Λ) sk 2(Λ) ... s1(Λ) − − then 1 k sk(Λ′)= sk(Λ) + dk(Λ). (5) −n Corollary 3.4. If Λ is realizable, then s (Λ′) 0. 2 ≥ Proof. A straightforward application of Theorem 3.3 yields n 2 1 2 s (Λ′)= − s (Λ) + s (Λ) 0. 2 n 2 n2 1 ≥ With this we see a particular case of the J-LL condition satisfied. Corollary 3.5. If Λ is a list, then s2(Λ) ns (Λ) (6) 1 ≤ 2 if and only if 2 s (Λ′) (n 1)s (Λ′). 1 ≤ − 2 Proof. Following (5), n 2 n 2 s2(Λ) ns (Λ) 0 − s (Λ) − s2(Λ) 1 ≤ 2 ⇐⇒ ≤ n 2 − n2 1 n 2 1 1 n 2 0 − s (Λ) + s2(Λ) s2(Λ) − s2(Λ) ⇐⇒ ≤ n 2 n2 1 − n2 1 − n2 1 n 2 1 n 1 0 − s (Λ) + s2(Λ) − s2(Λ) ⇐⇒ ≤ n 2 n2 1 − n2 1 n 2 1 n 1 2 0 (n 1) − s (Λ) + s2(Λ) − s (Λ) ⇐⇒ ≤ − n 2 n2 1 − n 1 2 0 (n 1)s (Λ′) s (Λ′) ⇐⇒ ≤ − 2 − 1 2 s (Λ′) (n 1)s (Λ′). ⇐⇒ 1 ≤ − 2 4 Inequality (6) is important for the realizability of lists with three elements [24, Theorem 2], as well as generalized Sule˘ımanova spectra [21], which we define in the sequel. A real list Λ is called a Sule˘ımanova spectrum if s1(Λ) 0 and Λ contains exactly one nonnegative entry.
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